Movements of controlled manipulators, e.g. industrial robots, are generally determined by a preestablished sequence of points in a multidimensional space. In the case of a multiaxial industrial robot each of these points incorporates the Cartesian coordinates of positions and coordinates with respect to the orientation in three-dimensional Euclidean space and optionally coordinates of auxiliary axes.
As a rule a robot movement is not rigidly predetermined, but instead merely has a number of fixed points, which are based on the operations to be performed, e.g. the taking up and setting down of workpieces, passing round obstacles, etc. The fixed points are taught to the robot control used for movement control purposes in a manual manner by starting and storing the corresponding poses and are then available as a point sequence with respect to which the robot movement to be executed must be oriented, i.e. all the fixed points clearly define the resulting path curve.
In order to obtain a handy, functional description in place of a point sequence and permit the continuity of the robot movement in all components of motion, the point sequence is generally interpolated. A functional description of the point sequence obtained by interpolation serves a number of purposes. Firstly the data quantity is compressed and secondly it is possible to obtain informations with respect to values not contained in the point sequence, e.g. by extrapolation. Finally, a higher evaluation rate occurs, because the evaluation of a function can often be implemented faster than a reference in a table.
In the industrial practice of movement control for interpolating the aforementioned point sequences use is frequently made of spline functions, which satisfy tailor-made demands, particularly with respect to the smoothness of the curve obtained, the continuity and differentiatability of the first derivative thereof and the continuity of the second derivative thereof. As the first and second derivatives of a path curve can be identified with the speed or acceleration of an object moved along the curve, said characteristics are of great importance for an efficient movement control of industrial robots.
A known method for the movement control of industrial robots using spline interpolations is described in an article by Horsch and Jüttler (Computer-Aided Design, vol. 30, pp 217-224, 1998). The contour of a robot movement is represented sectionwise by means of a suitable degree or order polynomial.
In the known method for the movement control of industrial robots by spline interpolation, it has proved disadvantageous that in sections of the point sequence to be interpolated in which changes to the orientation, the Cartesian position and possibly the position of the additional axes occur in widely differing form (e.g. minimum Cartesian changes with large orientation changes), undesired, uneconomic movement paths can occur. In particular, in such cases movement loops can occur, i.e. one or more components of the robot movement locally lead to an inefficient moving backwards and forwards, because the movement guidance in this specific section of the point sequence is dominated by significant changes in other components of motion. It is also considered disadvantage with per se fixed movement paths a subsequent pronounced change to individual components of motion, e.g. the subsequent programming of a strong reorientation, can have an effect on the behaviour of the other components of motion, so that e.g. as a consequence of the reorientation the Cartesian contour of the path also changes, which is unacceptable for the user.
For the orientation guidance of Cartesian movements, which are e.g. in the form of splines, frequent use is made of quaternions. Quaternions are generalized complex numbers in the form of a number quadruple, which can be represented as a real-value scalar part and complex-value three-vector. Quaternions are closely linked with the matrix representation of rotations and are therefore preferably used for the orientation guidance in computer animations and for robot control purposes. A point sequence of robot orientations is consequently represented by a four-dimensional spline in the quaternion space.
In order to obtain in this connection an optimum favourable, uniform parametrization of the orientation movement, the spline in the quaternion space should as far as possible be on the surface of the unit sphere. It is only in this way possible to achieve a uniform parametrization of the projection of the quaternion spline on the unit sphere, because on converting quaternions into rotation matrixes each quaternion qi must be standardized. Through the reduction of the four-dimensional quaternion spline to three degrees of freedom during projection on the unit sphere (standardization of the quaternion spline or the individual quaternions), the risk arises of a distorted parametrization of the standardized spline, which can in turn lead to an undesired movement behaviour of a robot controlled in this way.
In the known spline interpolation method there is a parametrization of the movement or spline by means of a common parameter, often called the motion parameter t and which is related to the time coordinate τ (cf. Horsch and Jüttler, top left on p 221). For a given point sequence Pi, i=1, . . . , n, the length of the parametrization interval [ti,ti+1] is generally considered to be proportional to the spacing of the associated points Pi, Pi+1. Into said spacing is introduced both the Cartesian distance and changes to the orientation and optionally auxiliary axes. However, it has been found that a parametrization by means of only a single parameter gives to the aforementioned disadvantages.
The problem of the invention is to further develop a method and device of the aforementioned type in such a way that there are no undesired, uneconomic path movements, such as loops or the like. In addition, movement or motion control is to take place in such a way that reorientation from point to point is at a minimum.